Remark that choosing \(\beta =\sqrt{V_Hn}\)  gives us a Sub-Gaussian concentration around \(T_H(P)\) , this is similar to the concentrations inequalities introduced in [1]} except that we concentrate around \(T_H(P)\)  instead of \(\mathbb {E}[X]\) . Remark also that the condition \(V_H\le \beta ^2/8\)  is rather weak because we already have \(V_H\le \beta ^2\) , the condition asks that there is enough weight in the interval \([-\beta ,\beta ]\) .
